Answer:
6, 12, 24, 48 and 96.
Step-by-step explanation:
We look for a geometric sequence with first term = 3:
nth term= a1 r^(n-1)
3 r^(n-1) = 192
r^(n-1) = 192 / 3 = 64
2 ^ 6 = 64 so r = 2.
Series is 3 ,6, 12, 24, 48, 96, 192.
So 5 geometric means would be 6, 12, 24, 48 and 96.
The geometric sequence will be 3, 6, 12, 24, 46, 96 and 192
Geometric mean can be defined as the average of the product of numbers which can either be Integer or fraction.
The nth term of a geometric progression is expressed as:
[tex]Tn = ar^{n-1}[/tex]
Tn is the nth term
a is the first term
r us the common ratio
n is the number of terms
Given that
a = 3
n = 192
Substitute into the formula to get r
[tex]192 = 3r^{n-1}\\r^{n-1} = 192/3\\r^{n-1} = 64\\[/tex]
Since n = 7 (total number of terms)
[tex]r^{7-1} = 64\\r^6 = 64\\r = 2\\[/tex]
Get the 2nd term
[tex]T_2 = 3r^{2-1} \\T_2 = 3(2)\\T_2 = 6\\[/tex]
Get the third term.
[tex]T3 = 3r^{3-1} \\T3 = 3(2)^2\\T3 = 12\\[/tex]
Get the fourth term
[tex]T4 = 3r^{4-1} \\T4 = 3(2)^3\\T4 = 24[/tex]
Get the fifth term
[tex]T5 = 3r^{5-1} \\T5 = 3(2)^4\\T5 = 48[/tex]
Get the 6th term
[tex]T6 = 3r^{6-1} \\T6 = 3(2)^5\\T6 = 96[/tex]
Hence the geometric sequence will be 3, 6, 12, 24, 46, 96 and 192
Learn more on geometric sequence here;
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